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Paper IPM / M / 437  


Abstract:  
For a finite group G, #Cent(G) denotes the number of centralizers of its elements. A group G is called ncentralizer if #Cent(G)=n, and primitive ncentralizer if #Cent(G) = #Cent([(G)/(Z(G))])=n.
In this paper we compute the number of distinct centralizers of some finite groups and investigate the structure of finite groups with exactly six distinct centralizers. We prove that if G is a 6centralizer group then [(G)/(Z(G))] ≅ D_{8},A_{4},Z_{2}×Z_{2}×Z_{2} or Z_{2}×Z_{2}×Z_{2} ×Z_{2}.
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